Relation with the Weil groupoid

Chapter VI: Constructible Weil Sheaves

6.5 Relation with the Weil groupoid

In this subsection, we relate lisse Weil sheaves with representations of the Weil groupoid. Throughout, we work with étale fundamental groups as opposed to their proétale variants in order to have Drinfeld’s lemma available, see Section 7.4. The two concepts differ in general, but agree for geometrically unibranch (for example, normal) Noetherian schemes, see (Bhatt and Scholze, 2015, Lemma 7.4.10).

For a Noetherian scheme 𝑋, let𝜋

1(𝑋) be theétale fundamental groupoid of 𝑋 as defined in (Revêtements étales et groupe fondamental (SGA 1)2003, Exposé V, §7 and §9). Its objects are geometric points of𝑋, and its morphisms are isomorphisms of fiber functors on the finite étale site of 𝑋. This is an essentially small category.

The automorphism group in 𝜋

1(𝑋) at a geometric point 𝑥 → 𝑋 is profinite. It is denoted𝜋

1(𝑋 , 𝑥)and called theétale fundamental groupof(𝑋 , 𝑥). If𝑋is connected, then the natural map𝐵𝜋

1(𝑋 , 𝑥) →𝜋

1(𝑋) is an equivalence for any𝑥 → 𝑋. If𝑋 is the disjoint sum of schemes𝑋_𝑖,𝑖 ∈ 𝐼, then𝜋

1(𝑋) is the disjoint sum of the𝜋

1(𝑋_𝑖), 𝑖 ∈𝐼. In this case, if𝑥→ 𝑋 factors through𝑋_𝑖, then𝜋

1(𝑋 , 𝑥)=𝜋

1(𝑋_𝑖, 𝑥). Definition 6.5.1. Let 𝑋

1, . . . , 𝑋_𝑛 be Noetherian schemes over F^𝑞, and write 𝑋 = 𝑋1×_F𝑞 . . .×_F𝑞 𝑋_𝑛. The Frobenius-Weil groupoid is the stacky quotient

FWeil(𝑋) =𝜋

1(𝑋

F)/⟨𝜙^Z

𝑋1

, . . . , 𝜙^Z

𝑋_𝑛⟩, (6.11)

where we use that the partial Frobenii𝜙_𝑋

𝑖 induce automorphisms on the finite étale site of 𝑋

F.

For 𝑛 = 1, we denote FWeil(𝑋) = Weil(𝑋). Even if 𝑋 is connected, its base change 𝑋

F might be disconnected in which case the action of 𝜙_𝑋 permutes some connected components. Therefore, fixing a geometric point of 𝑋

F is inconvenient, and the reason for us to work with fundamental groupoids as opposed to fundamental groups. The automorphism groups in Weil(𝑋)carry the structure of locally profinite groups: indeed, if 𝑋 is connected, then Weil(𝑋) is, for any choice of a geometric point𝑥 → 𝑋

F, equivalent to the classifying space of the Weil group Weil(𝑋 , 𝑥)from (Deligne, 1980, Définition 1.1.10).

Recall that this group sits in an exact sequence of topological groups 1→ 𝜋

1(𝑋

F, 𝑥) →Weil(𝑋 , 𝑥) → Weil(F/F^𝑞) ≃Z, (6.12) where 𝜋

1(𝑋

F, 𝑥) carries its profinite topology and Z the discrete topology. The topology on the morphism groups in Weil(𝑋) obtained in this way is independent

from the choice of 𝑥 → 𝑋

F. The image of Weil(𝑋 , 𝑥) → Zis the subgroup 𝑚Z where𝑚 is the degree of the largest finite subfield inΓ(𝑋 ,O𝑋). In particular, we have𝑚 = 1 if 𝑋

F is connected. Let us add that if𝑥 → 𝑋

F is fixed under𝜙_𝑋, then the action of 𝜙_𝑋 on 𝜋

1(𝑋

F, 𝑥) corresponds by virtue of the formula 𝜙^∗

𝑋 = (𝜙^∗

F)⁻¹ to the action of the geometric Frobenius, that is, the inverse of the𝑞-Frobenius in Weil(F/F^𝑞).

Likewise, for every 𝑛 ≥ 1, the stabilizers of the Frobenius-Weil groupoid are related to the partial Frobenius-Weil groups introduced in (V. G. Drinfeld, 1987, Proposition 6.1) and (V. Lafforgue, 2018, Remarque 8.18). In particular, there is an exact sequence

1→ 𝜋

1(𝑋

F, 𝑥) →FWeil(𝑋 , 𝑥) → Z^𝑛, for each geometric point 𝑥 → 𝑋

F. This gives FWeil(𝑋) the structure of a locally profinite groupoid.

Let Λ be either of the following coherent topological rings: a coherent discrete ring, an algebraic field extension𝐸 ⊃ Q^ℓ for some primeℓ, or its ring of integers O𝐸 ⊃ Z^ℓ. For a topological groupoid𝑊, we will denote by RepΛ(𝑊)the category of continuous representations of𝑊 with values in finitely presentedΛ-modules and by Rep^f_Λ^.p(𝑊) ⊂RepΛ(𝑊)its full subcategory of representations on finite projectiveΛ- modules. Here finitely presentedΛ-modules𝑀carry the quotient topology induced from the choice of any surjectionΛ^𝑛 → 𝑀,𝑛 ≥ 0 and the product topology onΛ^𝑛. Lemma 6.5.2. In the situation above, the category RepΛ(𝑊) is Λ_∗-linear and abelian. In particular, its full subcategoryRep^f_Λ^.p(𝑊) isΛ_∗-linear and additive.

Proof. Let𝑊

discbe the discrete groupoid underlying𝑊, and denote by RepΛ(𝑊

disc) the category of 𝑊

disc-representations on finitely presentedΛ-modules. Evidently, this category isΛ_∗-linear. It is abelian sinceΛis coherent. We claim that RepΛ(𝑊) ⊂ RepΛ(𝑊

disc) is aΛ_∗-linear full abelian subcategory. IfΛis discrete (and coherent), then every finitely presentedΛ-module carries the discrete topology and the claim is immediate, see also (Stacks, Tag 0A2H). ForΛ = 𝐸 ,O𝐸, one checks that every map of finitely presentedΛ-modules is continuous, every surjective map is a topological quotient and every injective map is a closed embedding. For the latter, we use that every finitely presentedΛ-module can be written as a countable filtered colimit of compact Hausdorff spaces along injections, and that every injection of compact Hausdorff spaces is a closed embedding. This implies the claim. □

We apply this for𝑊being either of the locally profinite groupoids𝜋

1(𝑋),𝜋

1(𝑋

F)or FWeil(𝑋). Note that restricting representations along𝜋

1(𝑋

F) →FWeil(𝑋)induces an equivalence ofΛ_∗-linear abelian categories

RepΛ FWeil(𝑋)

Fix RepΛ 𝜋

1(𝑋

F) , 𝜙_𝑋

1, . . . , 𝜙_𝑋

𝑛

, (6.13)

and similarly for theΛ_∗-linear additive category Rep^f_Λ^.p FWeil(𝑋) .

Definition 6.5.3. For an integer𝑛 ≥ 0, we writeD^{{−𝑛,𝑛}}_lis (𝑋 ,Λ) for the full subcate- gory ofD_lis(𝑋 ,Λ)of objects 𝑀such that𝑀 and its dual𝑀^∨lie in degrees [−𝑛, 𝑛] with respect to the t-structure onD(𝑋 ,Λ).

Lemma 6.5.4. In the situation above, there is a natural functor RepΛ FWeil(𝑋)

→D 𝑋^Weil

1 ×. . .× 𝑋^Weil

𝑛 ,Λ♥

, (6.14)

that is fully faithful. Moreover, the following properties hold if Λ is either finite discrete orΛ =O𝐸 for 𝐸 ⊃ Q^ℓ finite:

1. An object 𝑀 lies in the essential image of Equation (6.14) if and only if its underlying sheaf𝑀

Fis locally on(𝑋

F)_pro´etisomorphic to𝑁⊗_Λ∗Λ𝑋

F for some finitely presentedΛ∗-module𝑁.

2. The functor(6.14)restricts to an equivalence ofΛ_∗-linear additive categories Rep^f

.p

Λ FWeil(𝑋)

−→D^{0

,0} lis

𝑋^Weil

1 ×. . .×𝑋^Weil

𝑛 ,Λ .

3. IfΛ_∗ is regular (so thatΛis t-admissible, cf. Chapter 5 Item vi), then Equa- tion(6.14)restricts to an equivalence ofΛ_∗-linear abelian categories

RepΛ FWeil(𝑋)

−→ D_lis 𝑋^Weil

1 ×. . .× 𝑋^Weil

𝑛 ,Λ♥

.

If all 𝑋_𝑖, 𝑖 = 1, . . . , 𝑛are geometrically unibranch, then (1), (2) and (3) hold for general coherent topological ringsΛas above.

Proof. There is a canonical equivalence of topological groupoids𝜋

1(𝑋

F) 𝜋œ^pro´et

1 (𝑋

F) with the profinite completion of the proétale fundamental groupoid, see (Bhatt and Scholze, 2015, Lemma 7.4.3). It follows from (Bhatt and Scholze, 2015, Lemmas 7.4.5, 7.4.7) that restricting representations along𝜋^pro´et

1 (𝑋

F) → 𝜋

1(𝑋

F)induces full embeddings

RepΛ 𝜋

1(𝑋

F)

↩→RepΛ 𝜋^pro´et

1 (𝑋

F)

↩→D(𝑋

F,Λ)^♥, (6.15)

that are compatible with the action of𝜙_𝑋

𝑖 for all𝑖=1, . . . , 𝑛. So we obtain the fully faithful functor (6.14) by passing to fixed points, see (6.13), (6.7) and Lemma 4.2.2 (see also 4.2.3).

Part Item 1 describes the essential image of RepΛ 𝜋^pro´et

1 (𝑋

F)

↩→ D(𝑋

F,Λ)^♥. So ifΛis finite discrete or profinite, then the first functor in (6.15) is an equivalence, and we are done. Part Item 2 is immediate from Item 1, noting that an object in the essential image of Equation (6.15) is lisse if and only if its underlying module is finite projective. Likewise, part Item 3 is immediate from Item 1, using Item vii.

Here we need to exclude rings likeΛ = Z/ℓ²in order to have a t-structure on lisse sheaves.

Finally, if all 𝑋_𝑖 are geometrically unibranch, so is 𝑋

F which follows from the characterization (Stacks, Tag 0BQ4). In this case, we get𝜋

1(𝑋

F) 𝜋^pro´et

1 (𝑋

F) by (Bhatt and Scholze, 2015, Lemma 7.4.10). This finishes the proof. □